Delta-matroids and Vassiliev invariants

Lando, S. K.; Zhukov, Vyacheslav

doi:10.48550/arxiv.1602.00027

Cited by 2 publications

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“…In this section, we give general definitions related to delta-matroids and introduce the way the interlace polynomial can be generalized to delta-matroids. Our exposition follows [7]. Definition 2.…”

Section: Delta-matroidsmentioning

confidence: 99%

“…In this section we recall the 4-term relation for binary delta-matroids introduced in [7] and prove that the interlace polynomial of delta-matroids satisfies this relation.…”

Section: The 4-term Relation For the Interlace Polynomial For Delta-m...mentioning

confidence: 99%

“…The second Vassiliev move (handle sliding) was introduced and studied by I. Moffatt and E. Mphako-Banda in [8]. The first Vassliliev move (exchanging of handle ends) was defined by S. Lando and V. Zhukov in [7].…”

Section: The 4-term Relation For Delta-matroidsmentioning

confidence: 99%

“…Instead, one associates to an embedded graph another combinatorial structure, namely, its delta-matroid [2], and this delta-matroid is binary. As it was shown in [7], it is possible to define the four-term relation for binary delta-matroids in such a way that the mapping taking an embedded graph to its delta-matroid respects the corresponding four-term relations. This result raises the question whether known delta-matroid invariants respect the introduced relation.…”

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confidence: 99%

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The interlace polynomial of binary delta-matroids and link invariants

Kodaneva¹

2020

Preprint

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Section: Delta-matroidsmentioning

confidence: 99%

“…In this section we recall the 4-term relation for binary delta-matroids introduced in [7] and prove that the interlace polynomial of delta-matroids satisfies this relation.…”

Section: The 4-term Relation For the Interlace Polynomial For Delta-m...mentioning

confidence: 99%

Section: The 4-term Relation For Delta-matroidsmentioning

confidence: 99%

mentioning

confidence: 99%

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The interlace polynomial of binary delta-matroids and link invariants

Kodaneva¹

2020

Preprint

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“…Meanwhile, Lando and Zhukov in [10] constructed a Hopf algebra of binary delta-matroids, introduced four-term relations for them and constructed a quotient Hopf algebra modulo the four-term relations. The correspondence between delta-matroids and embedded graphs allows one to associate a weight system to a linear functional on the latter Hopf algebra.…”

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confidence: 99%

Lagrangian Subspaces, Delta-Matroids, and Four-Term Relations

Zhukov

2018

Funct Anal Its Appl

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Finite order invariants (Vassiliev invariants) of knots are expressed in terms of weight systems, that is, functions on chord diagrams satisfying the four-term relations. Weight systems have graph analogues, so-called 4-invariants of graphs, i.e. functions on graphs that satisfy the four-term relations for graphs. Each 4-invariant determines a weight system.The notion of weight system is naturally generalized for the case of embedded graphs with an arbitrary number of vertices. Such embedded graphs correspond to links; to each component of a link there corresponds a vertex of an embedded graph. Recently, two approaches have been suggested to extend the notion of 4-invariants of graphs to the case of combinatorial structures corresponding to embedded graphs with an arbitrary number of vertices. The first approach is due to V. Kleptsyn and E. Smirnov, who considered functions on Lagrangian subspaces in a 2n-dimensional space over F 2 endowed with a standard symplectic form and introduced four-term relations for them. On the other hand, the second approach, the one due to Zhukov and Lando, suggests four-term relations for functions on binary delta-matroids. In this paper, we prove that the two approaches are equivalent.Finite order invariants (Vassiliev invariants) of knots are expressed in terms of weight systems, that is, functions on chord diagrams satisfying four-term relations. The vector space over C spanned by chord diagrams considered modulo four-term relations is supplied with a Hopf algebra structure. The notion of weight system is naturally extended from functions on * The article was prepared within the framework of the Academic

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Delta-matroids and Vassiliev invariants

Cited by 2 publications

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The interlace polynomial of binary delta-matroids and link invariants

The interlace polynomial of binary delta-matroids and link invariants

Lagrangian Subspaces, Delta-Matroids, and Four-Term Relations

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